import math

def mystery(x):
    try:
        n = int(x)
    except:
        return False
    if n <= 1:
        return False
    # Check if n is prime
    for i in range(2, int(math.isqrt(n)) + 1):
        if n % i == 0:
            return False
    # Check if 2*n+1 is prime (Sophie Germain)
    m = 2 * n + 1
    for i in range(2, int(math.isqrt(m)) + 1):
        if m % i == 0:
            return False
    # Sum of digits
    s = sum(int(d) for d in str(n))
    # Check if s is a perfect cube
    cr = round(s ** (1/3))
    return cr ** 3 == s

import hashlib
import math

def mystery(x):
    s = str(x)
    if not s.isdigit():
        return False
    if len(s) != 5:
        return False
    if s == s[::-1]:
        return False
    n = int(s)
    if n <= 10000:
        return False
    if n % 2 == 0:
        return False
    h = hashlib.sha256(s.encode()).hexdigest()
    if h == "0" * 64:
        return False
    if n > 1:
        for i in range(2, int(math.isqrt(n)) + 1):
            if n % i == 0:
                break
        else:
            return False
    f = [1, 1]
    for i in range(2, 10):
        f.append(f[-1] * i)
    t = 0
    for c in s:
        t += f[int(c)]
    return t == n

import math
import hashlib

def mystery(x):
    # Convert input to integer; return False if not convertible.
    try:
        n = int(x)
    except (ValueError, TypeError):
        return False

    # --- Decoy checks and computations ---
    # These look important but are irrelevant for the final decision.
    if n <= 0:
        return False

    # Fake hash computation that doesn't affect logic.
    s = str(n).encode()
    dummy_hash = hashlib.sha256(s).hexdigest()
    dummy_sum = sum(int(d) for d in dummy_hash[:8] if d.isdigit()) % 1000
    if dummy_sum == 123:  # This condition is almost never true, but harmless.
        pass

    # Another decoy: modular check that is always true for valid n.
    if n % 1 != 0:
        return False

    # Bogus loop that scrambles some bits but never changes outcome.
    bits = n
    for _ in range(5):
        bits ^= (bits << 1) & 0xFFFFFFFF
    if bits & 0xFF == 0xDE:
        pass  # Do nothing

    # --- Real condition: n must be a Carmichael number with exactly 3 prime factors ---
    # Helper: simple primality test by trial division.
    def is_prime(num):
        if num < 2:
            return False
        for i in range(2, int(math.isqrt(num)) + 1):
            if num % i == 0:
                return False
        return True

    # Reject prime n.
    if is_prime(n):
        return False

    # Factor n by trial division.
    factors = []
    temp = n
    i = 2
    while i * i <= temp:
        if temp % i == 0:
            factors.append(i)
            while temp % i == 0:
                temp //= i
        i += 1 if i == 2 else 2  # Skip evens after 2.
    if temp > 1:
        factors.append(temp)

    # Must have exactly three distinct prime factors.
    if len(factors) != 3:
        return False

    # Ensure n is square-free (no repeated factor).
    for p in factors:
        if n % (p * p) == 0:
            return False

    # Carmichael condition: for each prime factor p, (n-1) % (p-1) == 0.
    for p in factors:
        if (n - 1) % (p - 1) != 0:
            return False

    return True

import hashlib

def mystery(x):
    try:
        s = str(x)
    except Exception:
        return False

    # Decoy: reject if contains 'xxx'
    if 'xxx' in s:
        return False

    # Decoy: checksum condition that eliminates some inputs
    checksum = sum(ord(c) for c in s) % 100
    if checksum == 50:
        return False

    # Decoy: MD5 computation that looks cryptographic but is unused
    md5_digest = hashlib.md5(s.encode()).hexdigest()
    if md5_digest.startswith('0000'):
        dummy = 1  # noise

    # Real condition: must be an 8-character lowercase string
    if len(s) != 8:
        return False
    if not s.islower():
        return False

    # Obfuscated transformations
    # Reverse the string
    rev = s[::-1]
    # Shift each character by its index in the reversed string
    shifted = ''.join(chr((ord(c) - 97 + i) % 26 + 97) for i, c in enumerate(rev))
    # Swap adjacent characters pairwise
    shifted_list = list(shifted)
    for i in range(0, len(shifted_list), 2):
        if i + 1 < len(shifted_list):
            shifted_list[i], shifted_list[i+1] = shifted_list[i+1], shifted_list[i]
    final = ''.join(shifted_list)

    # Target derived from the sample solution (ASCII values)
    target = ''.join(chr(c) for c in [104, 111, 108, 112, 102, 113, 109, 114])

    return final == target

import sys

def mystery(x):
    try:
        s = str(x)
    except Exception:
        return False

    # Decoy: reject if contains digit or uppercase
    if any(c.isdigit() or c.isupper() for c in s):
        return False

    # Decoy: meaningless checksum that rarely triggers
    checksum = sum(ord(c) * (i + 1) for i, c in enumerate(s)) % 1000
    if checksum == 777:
        return False  # very unlikely

    # Real constraint: must be exactly 8 lowercase letters
    if len(s) != 8 or not s.isalpha() or not s.islower():
        return False

    # Obfuscated transformation pipeline
    # Step 1: reverse
    t1 = s[::-1]

    # Step 2: position-dependent Caesar shift
    t2_list = []
    for i, ch in enumerate(t1):
        shift = i
        new_ord = (ord(ch) - 97 + shift) % 26 + 97
        t2_list.append(chr(new_ord))
    t2 = ''.join(t2_list)

    # Step 3: swap first half with second half
    half = len(t2) // 2
    t3 = t2[half:] + t2[:half]

    # At this point t3 should be "abcdefgh" for the correct input.

    # Decoy: fake hash computation (MD5) that is ignored
    import hashlib
    dummy_md5 = hashlib.md5(t3.encode()).hexdigest()
    if dummy_md5.startswith('dead'):
        pass  # never happens

    # Real check: custom 32‑bit polynomial hash of t3
    H = 0
    for c in t3:
        H = ((H << 5) - H + ord(c)) & 0xFFFFFFFF  # equivalent to H*31 + ord(c)
    target = 1259673732
    return H == target